---
title: Use morpheus package

output:
  pdf_document:
    number_sections: true
    toc_depth: 1
---

```{r setup,  results="hide", include=FALSE}
knitr::opts_chunk$set(echo = TRUE, include = TRUE,
  cache = TRUE, comment="", cache.lazy = FALSE,
  out.width = "100%", fig.align = "center")
```

## Introduction
<!--Tell that we try to learn classification parameters in a non-EM way, using algebric manipulations.-->

*morpheus* is a contributed R package which attempts to find the parameters of a
mixture of logistic classifiers.
When the data under study come from several groups that have different characteristics,
using mixture models is a very popular way to handle heterogeneity.
Thus, many algorithms were developed to deal with various mixtures models.
Most of them use likelihood methods or Bayesian methods that are likelihood dependent.
*flexmix* is an R package which implements these kinds of algorithms.

However, one problem of such methods is that they can converge to local maxima,
so several starting points must be explored.
Recently, spectral methods were developed to bypass EM algorithms and they were proved
able to recover the directions of the regression parameter
in models with known link function and random covariates (see [XX]).
Our package extends such moment methods using least squares to get estimators of the
whole parameters (with theoretical garantees, see [XX]).
Currently it can handle only binary output $-$ which is a common case.

## Model

Let $X\in \R^{d}$ be the vector of covariates and $Y\in \{0,1\}$ be the binary output.
A binary regression model assumes that for some link function $g$, the probability that
$Y=1$ conditionally to $X=x$ is given by $g(\langle \beta, x \rangle +b)$, where
$\beta\in \R^{d}$ is the vector of regression coefficients and $b\in\R$ is the intercept.
Popular examples of link functions are the logit link function where for any real $z$,
$g(z)=e^z/(1+e^z)$ and the probit link function where $g(z)=\Phi(z),$ with $\Phi$
the cumulative distribution function of the standard normal ${\cal N}(0,1)$.
Both are implemented in the package.

If now we want to modelise heterogeneous populations, let $K$ be the number of
populations and $\omega=(\omega_1,\cdots,\omega_K)$ their weights such that
$\omega_{j}\geq 0$, $j=1,\ldots,K$ and $\sum_{j=1}^{K}\omega{j}=1$.
Define, for $j=1,\ldots,K$, the regression coefficients in the $j$-th population
by $\beta_{j}\in\R^{d}$ and the intercept in the $j$-th population by
$b_{j}\in\R$. Let $\omega =(\omega_{1},\ldots,\omega_{K})$,
$b=(b_1,\cdots,b_K)$, $\beta=[\beta_{1} \vert \cdots,\vert \beta_K]$ the $d\times K$
matrix of regression coefficients and denote $\theta=(\omega,\beta,b)$.
The model of population mixture of binary regressions is given by:

\begin{equation}
\label{mixturemodel1}
\PP_{\theta}(Y=1\vert X=x)=\sum^{K}_{k=1}\omega_k g(<\beta_k,x>+b_k).
\end{equation}

## Algorithm, theoretical garantees

The algorithm uses spectral properties of some tensor matrices to estimate the model
parameters $\Theta = (\omega, \beta, b)$. Under rather mild conditions it can be
proved that the algorithm converges to the correct values (its speed is known too).
For more informations on that subject, however, please refer to our article [XX].
In this vignette let's rather focus on package usage.

## Usage
<!--We assume that the random variable $X$ has a Gaussian distribution.
We now focus on the situation where $X\sim \mathcal{N}(0,I_d)$, $I_d$ being the
identity $d\times d$ matrix. All results may be easily extended to the situation
where $X\sim \mathcal{N}(m,\Sigma)$, $m\in \R^{d}$, $\Sigma$ a positive and
symetric $d\times d$ matrix. ***** TODO: take this into account? -->

The two main functions are:
 * computeMu(), which estimates the parameters directions, and
 * optimParams(), which builds an object \code{o} to estimate all other parameters
   when calling \code{o$run()}, starting from the directions obtained by the
   previous function.
A third function is useful to run Monte-Carlo or bootstrap estimations using
different models in various contexts: multiRun(). We'll show example for all of them.

### Estimation of directions

In a real situation you would have (maybe after some pre-processing) the matrices
X and Y which contain vector inputs and binary output.
However, a function is provided in the package to generate such data following a
pre-defined law:

io <- generateSampleIO(n=10000, p=1/2, beta=matrix(c(1,0,0,1),ncol=2), b=c(0,0), link="probit")

n is the total number of samples (lines in X, number of elements in Y)
p is a vector of proportions, of size d-1 (because the last proportion is deduced from
  the others: p elements sums to 1) [TODO: omega or p?]
beta is the matrix of linear coefficients, as written above in the model.
b is the vector of intercepts (as in linear regression, and as in the model above)
link can be either "logit" or "probit", as mentioned earlier.

This function outputs a list containing in particular the matrices X and Y, allowing to
use the other functions (which all require either these, or the moments).

TODO: computeMu(), explain input/output

### Estimation of other parameters

TODO: just run optimParams$run(...)

### Monte-Carlo and bootstrap

TODO: show example comparison with flexmix, show plots.